As part of our attack on the protein-folding problem, we have been applying simple statistical-mechanical models to the study of secondary-structure formation in proteins. The backbone structure of polypeptide chains is described in terms of the conformations of individual structural units, which are pairs of flexible backbone dihedral angles. These units are viewed as having only two possible conformational states, the "native" state, corresponding to the conformation assumed by the unit in the native structure of the chain, and the "random-coil" state, which encompasses all non-native conformations. The structural states of a polypeptide chain are then identified by the sequence of native/random-coil states of these structural units. The stability of any such state is determined by the offsetting effects of entropy loss associated with fixing dihedral-angle pairs in the native conformation, and native non-bonding contacts between different parts of the chain. For a given chain these stabilizing contacts are identified by analyzing available X-ray crystallographic or NMR-derived structures of the corresponding protein. In our initial studies, the intra-chain contacts in a state could only connect parts of the chain separated by sufficiently long contiguous stretches of native dihedral-angle pairs. We have generalized this description to include the entropy of "loops" created by contacts between parts of the chain separated by stretches of non-native dihedral-angle pairs. Specified values for the entropy loss of fixing a single unit in the native conformation and for the overall strength of the intra-chain contacts are then used to compute a model partition function of the chain. The number of states that arises from complete enumeration of the possible combinations of dihedral-angle-pair states becomes intractably large for even a moderately long chain. However, we have developed methods for computing exact partition functions for model polypeptide chains of arbitrary length in certain cases. For the remaining cases we compute the partition function mostly using the "single-sequence" and "double-sequence" approximations--i.e., only considering states which have at most one and two contiguous stretches of native dihedral-angle pairs, respectively. These partition functions are used to compute the free energy for a given chain as a function of a single reaction coordinate given by, for example, the total number of native dihedral-angle pairs; this "reaction free-energy surface" provides the basis for modeling the equilibrium and kinetic folding properties of the chain. We have developed a procedure for "combinatorial modeling", which allow us to efficiently screen a very broad class of models, incorporating a wide variety of model assumptions--e.g., single-sequence vs. double-sequence vs. exact partition function calculation, different choices of reaction coordinate, different criteria for identifying intra-chain contacts, etc. We have found that this approach gives a useful qualitative picture of folding in simple proteins, and we are currently assessing the utility of this type of statistical-mechanical model for describing the folding of more complicated proteins and the effects of mutations on folding properties. In keeping with our interest in developing and exploiting new methods for probing the structural dynamics of macromolecules, we are developing new algorithms for the analysis of Laue (i.e., polychromatic illumination) diffraction data acquired in time-resolved X-ray crystallographic studies of proteins. These algorithms include efficient procedures for the assignment of Miller indices and integration of intensities of the observed reflections, and accurate atomic modeling of the resulting structure factors.